Electrodiffusive Formalism for Ion Concentration Dynamics in Excitable Cells and the Extracellular Space Surrounding Them

Fig. 1

A two domain-model for ion concentration dynamics in the intra- and extracellular space. The ICS is represented as a cylindrical cable (I), coated by ECS (E). The geometry is specified by three parameters, where a _Iis the cross section area of the cable, a _Eis the cross section area of the ECS and O _Mis the circumference of the cable. The concentration of ion species k is denoted [k]_nwhere n represents domain I or E. Ionic movement is described by the transmembrane flux density (j _kM) and the longitudinal flux densities due to electrical migration (j _kn^f) and diffusion (j _kn^d)

2.1 Particle Conservation

We consider the continuity equations for an ion species k with valence z _kin domains I and E:

$\displaystyle\begin{array}{rcl} \frac{\partial j_{kI}(x,t)} {\partial x} + \frac{O_{M}} {a_{I}} j_{kM}(x,t) + \frac{O_{M}} {a_{I}} j_{kI}^{in}(x,t) + \frac{\partial [k]_{I}(x,t)} {\partial t} & =& 0{}\end{array}$

(1)

$\displaystyle\begin{array}{rcl} \frac{\partial j_{kE}(x,t)} {\partial x} -\frac{O_{M}} {a_{E}} j_{kM}(x,t) + \frac{O_{M}} {a_{E}} j_{kE}^{in}(x,t) + \frac{\partial [k]_{E}(x,t)} {\partial t} & =& 0,{}\end{array}$

(2)

with the sealed-end boundary conditions (n = I or E):

$\displaystyle{ j_{kn}(0,t) = j_{kn}(l,t) = 0. }$

(3)

Here a _Iand a _Eare the cross sections of the ICS and ECS, respectively, and O _Mis the circumference of the membrane. The longitudinal flux densities are given by the generalized Nernst-Planck equation (to keep notation short, we skip the functional arguments (x, t) from here on):

$\displaystyle{ j_{kn} = -\frac{D_{k}} {\lambda _{n}^{2}} \frac{\partial [k]_{n}} {\partial x} -\frac{D_{k}z_{k}} {\lambda _{n}^{2}\psi } [k]_{n}\frac{\partial v_{n}} {\partial x}, }$

(4)

where the first term on the right represents the diffusive flux density (j _kn^d) and the last term is the flux density due to ionic migration in the electrical field (j _kn^f). The effective diffusion constant $D_{k}^{{\ast}} = D_{k}/\lambda _{n}^{2}$ is composed of the diffusion constant D _kin dilute solutions and the tortuosity factor λ _n, which summarizes the hindrance imposed by the cellular structures [2, 10]. We use $\psi = RT/F$ , where R is the gas constant, T the absolute temperature, and F is Faraday’s constant. The formalism we derive is general to the transmembrane flux density (j _kM), as long as j _kMis a local function of v _M, ionic concentrations in I and E, and possibly some additional local state variables. The formalism can be combined with any external input (j _knⁱⁿ) which fulfills the constraint:

$\displaystyle{ \sum _{k}z_{k}j_{kE}^{in} = -\sum _{ k}z_{k}j_{kI}^{in}, }$

(5)

as we shall explain later.

With N ion species, Eqs. 1 and 2 (with j _kn, j _kMand j _knⁱⁿas described above) represent a system of 2N + 3 variables which are functions of x and t. These are the 2N concentration variables ([k]_nfor $k = 1,2,\ldots N$ and n = E, I), and the three additional variables ( $v_{M},\partial v_{I}/\partial x$ and $\partial v_{E}/\partial x$ ) occurring in the expressions for the flux densities. We now seek to express $v_{M},\partial v_{I}/\partial x$ and $\partial v_{E}/\partial x$ as functions of ionic concentrations, so that Eqs. 1 and 2 constitute a fully specified (and numerically solvable) system of equations.

2.2 Voltage Expressions

To reduce the number of independent variables to the 2N state variables ([k]_n) we use three additional constraints:

$\displaystyle{ v_{M} = \frac{a_{I}} {C_{M}O_{M}}\rho _{I} }$

(6)

$\displaystyle{ \frac{a_{I}} {C_{M}O_{M}}\rho _{I} = - \frac{a_{E}} {C_{M}O_{M}}\rho _{E}. }$

(7)

$\displaystyle{ v_{M} = v_{I} - v_{E} \Rightarrow \frac{\partial v_{M}} {\partial x} = \frac{\partial v_{I}} {\partial x} -\frac{\partial v_{E}} {\partial x} }$

(8)

Equation 6 is the assumption that the membrane is a parallel plate capacitor. Then v _Mis determined by the density of charge on the inside of the membrane, which in turn is determined by the ionic concentrations:

$\displaystyle{ \rho _{n} = F\sum _{k}z_{k}[k]_{n} +\rho _{sn}. }$

(9)

For practical purposes, we have included a density of static charges (ρ _sn) in Eqs. 6 and 7, representing contributions from ions that are not considered in the conservation equations. If the set [k]_ninclude all present species of ions, then ρ _sn = 0.

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